There are many flights available to Flesland airport (code BGO) just outside of Bergen. After leaving the terminal building, take one of the frequently departing buses (every 15 minutes) to the city centre. The bus ride can be paid for by card and takes about half an hour.
We recommend to avoid flights from abroad via Oslo, since Norwegian customs regulations require you to collect your luggage at the first Norwegian airport you arrive at, even if it's not your final destination. In Oslo, you would thus have to wait at the baggage claim, carry your luggage through customs, leave the secure area, check the baggage in again at a counter, make your way through security, and find your gate in the domestic leg of the airport. This process may very well take more than an hour.
In case you have extra time to spare, a more picturesque option is to take the beautiful train ride from Oslo to Bergen. It takes about 8 hours and goes all the way up to 1237 m ASL on the Hardangervidda plateau. This video gives a taste of the experience.
